Finite Group Databasebeta

← C3×#10592518 #13162829

Subgroups — C3×#10592518 #13162829

Order 1728.

912 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic
4C3×#64102C211yesmaximal, characteristic, complemented
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63641C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592518C311yesmaximal, characteristic, direct product factor
9C3×#4819—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3755—22no—
17C3×#3752—21no—
18C3×#3752—21no—
19C3×#3752—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3745—22no—
23#64156C2×C311yescharacteristic, complemented
24C2×#3755C2×C311yescharacteristic, complemented
25C2×#3745C2×C311yescharacteristic
26#64102C2×C311yescharacteristic, complemented
27C3×#947—34no—
28C2×C3×#215—34no—
29C2×C3×D24—34no—
30C3×#893—34no—
31C32×#103—24no—
32C32×#103D314yescomplemented
33C32×#103C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—34no—
37C3×#809—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)D412yes—
51C3×C4×C3⋊D3—42no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#463—91no—
55#4819—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×C3×D12—34no—
63C2×C3×D12—34no—
64C2×C3×C4×D3—64no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C22×C311yescharacteristic
68C32×C4⋊C4—24no—
69C32×C4⋊C4C2×D314yes—
70C32×C4⋊C4C22×C311yescharacteristic
71#3755—22no—
72#3752—21no—
73#3752—21no—
74#3752—21no—
75C4×C3⋊Dic3C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77#3745—22no—
78C3×#215—68no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×C4×Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C3×D24—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C2×C3×SD16—91no—
99C2×C3×D8—91no—
100C3×#124—91no—
101C3×C4×D4—91no—
102#947—34no—
103C2×#215—34no—
104C2×D24—34no—
105#893—34no—
106C3×C4⋊C8—91no—
107C3×#103—24no—
108C3×#103C3×D314yescomplemented
109C3×#103C3⋊D311yescharacteristic, complemented
110C3×#102—91no—
111Dic3⋊C8—34no—
112#809—34no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3—64no—
117C32×Q8—28no—
118C32×Q8—24no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—24no—
122C32×Q8—44no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4—44no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C3⋊Dic3C3×D411yescharacteristic
133C2×C3⋊Dic3C3×D411yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)C3×D412yes—
138C4×C3⋊D3—42no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C3×D12—38no—
142C3×D12—64no—
143C3×D12—64no—
144C3×D12—64no—
145C3×C4×D3—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—44no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—64no—
151#463—91noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3×C3⋊D3—42no—
156C3×C3⋊D3—81no—
157C2×C3×Q8—24no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—91no—
161C2×C3×D4—91no—
162C22×C3×C4—181no—
163C2×D12—34no—
164C2×D12—34no—
165C2×C4×D3—64no—
166C3×SD16—182no—
167C3×D8—182no—
168C2×C3×C8—24no—
169C2×C3×C8—91no—
170C2×C3×C8C2×C3×D314yes—
171C2×C3×C8C2×C3⋊D311yescharacteristic
172C3×C4⋊C4—24no—
173C3×C4⋊C4C2×C3×D314yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C3×C22⋊C4—181no—
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178C3×C42—91no—
179#215—68no—
180(C2×D3)⋊C4—64no—
181(C2×D3)⋊C4—64no—
182(C2×D3)⋊C4—64no—
183C4×Dic3—34no—
184C2×C3⋊C8—34no—
185D24—68no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C22×C32C2×C3×D411yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3×D3—68no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—48no—
204C32×C4—44no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4—44no—
208C32×C4—84no—
209C32×C4#23514yes—
210C32×C4D3×D414yes—
211C32×C4C2×C3×D411yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—21no—
214C3⋊Dic3—21no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C2×SD16—91no—
218C2×D8—91no—
219#124—91no—
220C4×D4—91no—
221C4⋊C8—91no—
222#103C3×C3⋊D311yescharacteristic, complemented
223#102—91no—
224C33#46311yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C23×C3—181no—
227C22×D3—64no—
228C22×D3—64no—
229C3×Q8—28no—
230C3×Q8—24no—
231C3×Q8—24no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—44no—
235C3×D4—92no—
236C3×D4—181no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—362no—
249C2×C3×C4C22×C3×D314yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251C2×Dic3—34no—
252C2×Dic3—34no—
253D12—38no—
254D12—64no—
255D12—64no—
256D12—64no—
257C4×D3—128no—
258C3×C8—24no—
259C3×C8—21no—
260C3×C8—44no—
261C3×C8—181no—
262C3⋊C8—64no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32#88414yes—
267C2×C32#91414yes—
268C2×C32#91914yes—
269C2×C32C3×#12111yescharacteristic
270C2×C32C3×#13511yescharacteristic
271C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
272C3⋊D3—42no—
273C3⋊D3—81no—
274C3×D3—128no—
275C3×D3—244no—
276C2×Q8C2×C3×C3⋊D311yescharacteristic
277C2×D4—91no—
278C2×D4—91no—
279C22×C4—181no—
280SD16—182no—
281D8—182no—
282C2×C8—91no—
283C2×C8C2×C3×C3⋊D311yescharacteristic
284C4⋊C4C2×C3×C3⋊D311yescharacteristic
285C22⋊C4—181no—
286C22⋊C4—181no—
287C22⋊C4—181no—
288C42—91no—
289C22×C3—24no—
290C22×C3—182no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3C3×D3×D414yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—68no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—124no—
302C2×D3—124no—
303C2×D3—124no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—48no—
307C3×C4—44no—
308C3×C4—44no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—84no—
312C3×C4—181no—
313C3×C4—181no—
314C3×C4C3×#23514yes—
315C3×C4C3×D3×D414yes—
316C3×C4#382811yescharacteristic
317C3×C4D4×C3⋊D311yescharacteristic
318Dic3—64no—
319Dic3—64no—
320C32—24no—
321C32#481914yescomplemented
322C32C3×#46311yescharacteristic, complemented
323C23—181no—
324C23—181no—
325Q8—21no—
326Q8—21no—
327D4—92no—
328D4—181no—
329D4—181no—
330D4—181no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—91no—
334C2×C4—91no—
335C2×C4—362no—
336C2×C4C22×C3×C3⋊D311yescharacteristic
337C8—21no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—362no—
343C2×C3—721no—
344C2×C3C3×#88414yes—
345C2×C3C3×#91414yes—
346C2×C3C3×#91914yes—
347C2×C3#6409311yescharacteristic
348C2×C3#6412311yescharacteristic
349C2×C3#6412811yescharacteristic
350D3—128no—
351D3—244no—
352C22—182no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22C3×D4×C3⋊D311yescharacteristic
359C4—41no—
360C4—41no—
361C4—181no—
362C4—181no—
363C4C3×#382811yescharacteristic
364C4C3×D4×C3⋊D311yescharacteristic
365C3—24no—
366C3C3×#481914yescomplemented
367C3#1059251811yescharacteristic, direct product factor
368C2—362no—
369C2—721no—
370C2C3×#6409311yescharacteristic
371C2C3×#6412311yescharacteristic
372C2C3×#6412811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C3×#3745C2
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C3×#3755C2semidirect
4. C3×#64102C2semidirect
5. C33×#103C2semidirect
6. C3×#63641C2semidirect
7. C3×#63636C2semidirect
8. #10592518C3direct (N × H)
23. #64156C2×C3semidirect
24. C2×#3755C2×C3semidirect
26. #64102C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
34. #63641C2×C3semidirect
35. #63636C2×C3semidirect
108. C3×#103C3×D3semidirect
109. C3×#103C3⋊D3semidirect
222. #103C3×C3⋊D3semidirect
224. C33#463semidirect
321. C32#4819semidirect
322. C32C3×#463semidirect
366. C3C3×#4819semidirect
367. C3#10592518direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#641562
C2×C3×#37552
C2×C3×#37452
C3×#641022
C33×#1032
C3×#636412
C3×#636362
#105925183
C3×#48193